Age of Information Performance of Multiaccess Strategies ...

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244 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 21, NO. 3, JUNE 2019

Age of Information Performance of MultiaccessStrategies with Packet Management

Antzela Kosta, Nikolaos Pappas, Anthony Ephremides, and Vangelis Angelakis

Abstract: We consider a system consisting of N source nodes com-municating with a common receiver. Each source node has a bufferof infinite capacity to store incoming bursty traffic in the form ofstatus updates transmitted in packets, which should maintain thestatus information at the receiver fresh. Packets waiting for trans-mission can be discarded to avoid wasting network resources forthe transmission of stale information. We investigate the age of in-formation (AoI) performance of the system under scheduled andrandom access. Moreover, we present analysis of the AoI withand without packet management at the transmission queue of thesource nodes, where as packet management we consider the ca-pability to replace unserved packets at the queue whenever newerones arrive. Finally, we provide simulation results that illustratethe impact of the network operating parameters on the age perfor-mance of the different access protocols.

Index Terms: Age of information, multiple-access channels, packetmanagement, performance analysis, queueing theory, real time sys-tems.

I. INTRODUCTION

FUTURE networks should support applications with hetero-geneous QoS requirements, where critical performance in-

dicators are the end-to-end delay, the throughput, the energy ef-ficiency, and the service reliability. The concept of age of infor-mation (AoI) was introduced in [1], [2] to quantify the freshnessof the knowledge we have about the status of a remote system.The age captures the time elapsed since the last received mes-sage containing update information was generated. The noveltyof this metric to characterize the freshness of information in acommunication system differentiates it from other conventionalmetrics such as delay and connects it with emerging real-timewireless applications.

Maintaining data freshness is a requirement in numerous ap-plications like wireless sensor networks (WSN) for healthcare

Manuscript received November 29, 2018; approved for publication May 20,2019.

The research leading to these results has been partially funded by the Euro-pean Union’s Horizon 2020 research and innovation programme under the MarieSklodowska-Curie Grant Agreement No. 642743 (WiVi-2020). The work of N.Pappas is supported in part by ELLIIT and the Center for Industrial InformationTechnology (CENIIT). The work of A. Ephremides is supported by the U.S. Of-fice of Naval Research under Grant ONR 5-280542, the U.S. National ScienceFoundation under Grants CIF 5-243150, Nets 5-245770, and CIF 5-231912, andthe Swedish Research Council (VR).

A. Kosta, N. Pappas, and V. Angelakis are with the Department of Scienceand Technology, Linköping University, Campus Norrköping, 60 174, Sweden,email: {antzela.kosta, nikolaos.pappas, vangelis.angelakis}@liu.se.

A. Ephremides is with the Electrical and Computer Engineering Depart-ment, University of Maryland, College Park, MD 20742, USA, email:[email protected].

A. Kosta is the corresponding author.Digital Object Identifier: 10.1109/JCN.2019.000039

and environmental monitoring, active data warehousing, energyharvesting [3]–[8], web caching [9]–[12], real time databases, adhoc networks [13], wireless smart camera networks [14], UAV-assisted IoT networks [15], [16], broadcast and multicast wire-less networks [17]–[20], etc. Moreover, in the field of adap-tive transmission significant efficiency gains can be obtained byadaptive signaling strategies. However, this feedback scheme isconstrained by the acquisition of timely channel state informa-tion (CSI) [21]–[24].

The first attempts to address the AoI of a source at the destina-tion of a status update transmission system were made throughsimple queueing models. In [25], three simple models werestudied, the M/M/1, the M/D/1, and the D/M/1, under the first-come-first-served (FCFS) discipline. Alternative measures ofstale information that are by-products of AoI are studied for theM/M/1 queue in [26]. An expansion of the basic model that in-cludes multiple sources sharing a common queue is consideredin [27]–[29]. The analysis therein illustrated how combiningmultiple sources in a common queue is more efficient in termsof the average AoI of each source, than serving them separately.

Moving to different system characteristics, in [30] the authorsconsider different systems with either plentiful or limited net-work resources (servers). Under this assumption, a more dy-namic feature of networks is considered, that is, packets trav-eling over a network might reach the destination through nu-merous alternative paths thus the delay of each packet mightdiffer. In this context, the performance of the M/M/1, M/M/2,and M/M/∞ queues is provided, and the tradeoff between AoIand the waste of network resources in terms of non-informativepackets as the number of servers varies, is demonstrated.

Two efficient ways to avoid congestion in networks are packetmanagement techniques and admission control, since they canmanage the traffic entering them. Packet management by drop-ping or replacing packets was investigated in [31], [32] wherethe M/M/1/1, M/M/1/2, and M/M/1/2* queues are considered. Akey outcome was that packet management can promote smalleraverage AoI, when compared to schemes without replacementand the same number of servers. The last-come-first-served(LCFS) queue discipline differs from packet management in thatpackets are not dropped if an infinite buffer is considered. Al-lowing newly generated status updates to surpass older statusupdates, with and without the use of preemption, was studied in[33]–[39].

Furthermore, a diversity of additional resource sharing fea-tures of a communication system have been studied in rela-tion to AoI. Transmission scheduling is considered in [40]–[48]where centralized and decentralized scheduling policies for AoIminimization, under general interference constraints and timevarying channels, are proposed. The proposed scheduling al-

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KOSTA et al.: AGE OF INFORMATION PERFORMANCE OF MULTIACCESS ... 245

gorithms have low complexity with strong AoI performancesover stochastic information arrivals. In [49] the authors considerscheduled access and slotted ALOHA-like random access, how-ever the queueing aspect along with random access is not cap-tured. Throughput and AoI performance in a cognitive sharedaccess network with queueing analysis has been studied in [50].Additional references can be found in the survey [51].

A. Contribution

In this work, we focus on the AoI performance of a networkconsisting of N source nodes communicating with a commonreceiver. Each source node has a buffer of infinite capacity tostore incoming bursty traffic in the form of packets which shouldkeep the receiver timely updated. We consider that the sourcenodes can discard packets waiting for transmission, in a processthat is referred to as packet management. We present analysisof the time average AoI with and without packet management atthe transmission queue of the source nodes. We investigate threedifferent policies to access the common medium (i) a round-robin scheduler (ii) a work-conserving scheduler (iii) randomaccess. To incorporate the effect of channel fading and networkpath diversity in such a system we provide simulation resultsthat illustrate the impact of network operating parameters on theperformance of the different access protocols. The network pathdiversity refers to the transmission of packets over multiple al-ternate paths.

II. SYSTEM MODEL

We consider a wireless network consisting ofN source nodescommunicating with a common receiver. Each node has a bufferof infinite capacity to store incoming packets. These packets arethen sent through error-prone channels to the destination d, asshown in Fig. 1. Packets have equal length and time is dividedinto slots such that the transmission time of a packet from thebuffer to the destination is equal to one slot. Each such packetis said to provide a status update and these two terms are usedinterchangeably. The status updates arrivals are modeled by in-dependent and identically distributed (i.i.d.) Bernoulli processeswith average probabilities λi ∈ (0, 1), for i = 1, · · ·, N . Theprobability distribution of time until successful delivery is as-sumed to be geometric with mean 1/µi slots, for the ith node,where µi is referred as the service rate of the ith node.

We consider two different queue disciplines: without and withpacket replacement. The first, assumes that all packets need tobe delivered to the destination regardless of the freshness of thestatus update information. We note that the motivation behindthis discipline is in terms of the reconstructability of the trans-mitted information (that can also be related with estimation andprediction theory aspects) that is beyond the scope of this work.The second discipline assumes that a packet which arrives whileanother packet is being served may be kept in the queue wait-ing for transmission. However, the packets waiting for trans-mission are replaced by newly generated packets of the samesource. We denote this discipline by replacement and the pro-cess of discarding the packets from the queue is referred to aspacket management. The packet management is expected to im-prove the performance of the system with respect to the staleness

λ2s2 d

λ1s1

λNsN

µ1

µN

µ2

Fig. 1. Status updates over a multiaccess network.

of the transmitted information. Nevertheless, note that this is anon-conventional queueing model, for which some of the classicresults from queueing theory, such as Little’s law, do not apply[52].

Status updates depart from the queues either in a perfectlyscheduled or a random fashion. We consider three different poli-cies to access the common medium.• Round-robin: The scheduler assigns time slots to each node

in equal portions and in fixed circular order.• Work-conserving: The scheduler makes probabilistic deci-

sions in each time slot, among the nodes that have a packet atthe transmission queue.

• Random: The nodes attempt to transmit the packet at the headof the queue with a given probability qi colliding with eachother.

These policies will be presented, evaluated, and compared in thenext sections in terms of their AoI performance.

III. AGE OF INFORMATION ANALYSIS

To derive the time average AoI of the system we start by char-acterizing AoI in terms of random variables that capture the ageevolution at the receiver. The age at the receiver depends on thepacket receptions and the delay imposed by the network to thesepackets. Then, expectations of the random variables are calcu-lated for each of the queue disciplines separately. In the nextsection, we evaluate AoI for the proposed access policies wherethe exact service rate at the queues is incorporated.

Consider that the jth status update of node i is generatedat time tij , delivered through the transmission system, andreceived by the destination at time t

′

ij . Then, we denote byTij = t′ij − tij the system time of update j of the ith node.This corresponds to the sum of the queueing time and the queueservice time. The interarrival time of update j of node i is de-fined as the random variable Yij = tij − ti(j−1). Finally, letZij = t′ij − t′i(j−1) be the random variable denoting the timebetween the reception of status update (j − 1) and j of node i.

The AoI of each source node at the destination is defined asthe random process ∆t = t − u(t), where u(t) is the times-tamp of the most recently received update from that source. An


t

∆t

∆0

t0 t1 t2 t3 t4 tn−1 tnt′1 t′2 t′3 t′n

J1 J2 J4 Jn

J

J3

Y2 T2 Yn Tn

Z2 Z3

Fig. 2. Example of age evolution of node i at the receiver.

illustrative example of the evolution of the age of information ofsource i in time is shown in Fig. 2. Without loss of generality,we assume that the observation of the system starts at t = 0.At that time the queues are empty, and the AoI of the ith nodeat the destination is ∆0. In the time intervals [t

′

i(j−1), t′

ij ], ∀j,the AoI increases in a stair-step fashion due to the absence ofupdates from node i at the destination. Upon reception of a sta-tus update from node i the AoI of that node is reset to a smallervalue that is equal to the delay that the packet experienced.

Ensuring the average AoI of the ith node is small correspondsto maintaining information about the status of the node at thedestination fresh. For presentation clarity, from now on we dropthe index denoting the source and focus on the packet index.Given an age process ∆t and assuming ergodicity, the averageage can be calculated using a sample average that converges toits corresponding stochastic average. For an interval of observa-tion (0, T ), the time average age of node i is

∆T =1

T

N(T )∑t=0

∆t, (1)

when we assume that the observation interval ends with the ser-vice completion ofN(T ) samples. The summation in (1) can becalculated as the area under ∆t. Then, the time average age canbe rewritten as a sum of disjoint geometric parts. Starting fromt = 0, the area is decomposed into the area J1, the areas Jj forj = 2, 3, · · ·, N(T ), and the area of width Tn that we denote byJ . Then, the decomposition of ∆T yields

∆T =1

T

J1 + J +

N(T )∑j=2

Jj

=J1 + J

T+N(T )− 1

T1

N(T )− 1

N(T )∑j=2

Jj . (2)

The time average ∆T tends to the ensemble average age as T →

∞, i.e.,∆ = lim

T→∞∆T .

1 (3)

Note that the term (J1 + J)/T goes to zero as T grows and alsolet

λ = limT→∞

N(T )

T(4)

be the steady state rate of status updates generation. Further-more, using the definitions of the interarrival and system times,we can write the areas Jj as

Jj =

Yj+Tj∑m=1

m−Tj∑m=1

m

=1

2(Yj + Tj)(Yj + Tj + 1)− 1

2Tj(Tj + 1)

= YjTj + Y 2j /2 + Yj/2. (5)

Then, substituting (2), (4), and (5), to (3) the average age ofinformation of the ith node is given by

∆ = λ

(E[Y T ] +

E[Y 2]

2+

E[Y ]

2

), (6)

where E[·] is the expectation operator. The expression obtainedin (6) differs from the expression obtained in [25] for the contin-uous time setup of the problem by an additional term E[Y ]/2.

Alternatively, we can express the areas Jj with respect to therandom variables Zj , as follows

Jj =

Tj−1+Zj∑m=1

m−Tj∑m=1

m

=1

2(Tj−1 + Zj)(Tj−1 + Zj + 1)− 1

2Tj(Tj + 1), (7)

and utilize the fact that when the system reaches steady stateTj−1 and Tj are identically distributed. We use E[T ] to repre-sent the expected value of Tj for an arbitrary j. Taking expecta-tions of both sides gives

E[J ] = E[ZT ] + E[Z2]/2 + E[Z]/2. (8)

Then, substituting (2), (4), and (8), to (3) the average age ofinformation of the ith node is given by

∆ = λ

(E[ZT ] +

E[Z2]

2+

E[Z]

2

). (9)

In what follows, we analyze the steady-state age of informa-tion without and with packet management at the transmissionqueues.

A. Geo/Geo/1 Queue

First, we derive the average AoI in (6) of the ith node withoutpacket management, at the destination. The interarrival times Yjare i.i.d. sequences that follow a geometric distribution therefore

1We assume that the existence of the limit is guaranteed by the stability of thequeues.


0 1 2 3 · · ·(1− λ)

λ λ(1− µ) λ(1− µ) λ(1− µ)

µ(1− λ)µ(1− λ)µ(1− λ)µ(1− λ)

1− r − s 1− r − s 1− r − s

Fig. 3. The DTMC which models the Geo/Geo/1 queue evolution at node i.

we know that

E[Yj ] =1

λ, E[Y 2

j ] =2− λλ2

. (10)

Then, the only unknown term for the calculation of the averageage is the expectation E[Y T ]. The system time of update j isTj = Wj + Sj , where Wj and Sj are the waiting time andservice time of update j, respectively. Since the service timesSj are independent of the interarrival times Yj , we can write

E[YjTj ] = E[Yj(Wj + Sj)] = E[YjWj ] + E[Yj ]E[Sj ], (11)

where E[Sj ] = 1/µ. Moreover, we can express the waiting timeof update j as the remaining system time of the previous updateminus the elapsed time between the generation of updates (j−1)and j, i.e.,

Wj = (Tj−1 − Yj)+. (12)

Note that if the queue is empty then Wj = 0. Also notethat when the system reaches steady state the system times arestochastically identical, i.e., T =st Tj−1 =st Tj .

In addition, the queue of the ith node can be describedthrough a discrete-time Markov chain (DTMC), where eachstate represents the number of packets in the queue.

Lemma 1: From the DTMC described in Fig. 3 we obtainthe following steady state probabilities

πn = ρn−1π1, n ≥ 1, and π0 =µ(1− λ)

λπ1, (13)

where ρ = λ(1−µ)µ(1−λ) , π1 = λ(1−ρ)

µ , r = λ(1 − µ), and s =

µ(1− λ).

To derive the probability mass function (pmf) of the systemtime T , we use the fact that the sum of N geometric randomvariables where N is geometrically distributed is also geometri-cally distributed, according to the convolution property of theirgenerating functions [53]. Let Sj , j = 1, 2, · · · be independentand identically distributed geometric random variables with pa-rameter µ. If an arriving packet sees N packets in the system,then, the system time of that packet, using the memoryless prop-erty, can be written as the random sum T = S1 + · · ·+ SN . Tocalculate the probability generating function of T we conditiononN = nwhich occurs with probability (1−ρ)ρn−1 and obtain

GT (z) =

∞∑n=1

(µz

1− (1− µ)z

)n(1− ρ)ρn−1

=µ(1− ρ)z

1− (1− µ(1− ρ))z. (14)

This implies that the system time pmf is given by

fT (t) = µ(1− ρ)(1− µ+ µρ)t−1. (15)

As a result, T follows a geometric distribution with parameterµ(1− ρ). An alternative approach that uses moment generatingfunctions can also be found in [43].

Now we are able to compute the conditional expectation ofthe waiting time Wj given Yj = y as

E[Wj |Yj = y] = E[(Tj−1 − y)+|Yj = y] = E[(T − y)+]

=

∞∑t=y

(t− y)fT (t) =(1− µ+ µρ)y

µ(1− ρ). (16)

Then, the expectation E[WjYj ] is obtained as

E[WjYj ] =

∞∑y=0

y E[Wj |Yj = y] fYj(y)

=λ(1− µ+ µρ)

µ(1− ρ)(λ+ µ− λµ− µρ+ λµρ)2. (17)

Substituting ρ = λ(1−µ)µ(1−λ) to (17) and after some algebra we ob-

tain

E[WjYj ] =λ(1− µ)

(µ− λ)µ2. (18)

From (18), (11), and (6), the average AoI of the ith node is ob-tained as

∆Geo/Geo/1 =1

λ+

1− λµ− λ

− λ

µ2+λ

µ. (19)

In order to find the optimal value of λ that minimizes the av-erage AoI we proceed as follows. We differentiate (19) withrespect to λ to obtain ∂∆/∂λ. By setting ∂∆/∂λ = 0 we canobtain the value of λ that minimizes the AoI and satisfies theequation λ4(µ− 1)− 2λ3(µ− 1)µ− λ2µ2 + 2λµ3 − µ4 = 0.Trivially one can see that ∆ is a convex function of λ for a givenservice rate µ, if λ < µ is not violated, by taking the secondderivative ∂2∆/∂λ2.

B. Queue with Replacement

Next, the queue with replacement at the ith node can be de-scribed as a three-state discrete-time Markov chain where eachstate represents an empty system, a single packet receiving ser-vice, or a packet in the queue waiting for a packet in the server,respectively, as in [31]. The packet replacement does not affectthe number of packets in the system since a newly generatedpacket discards the packet waiting in the queue, if any.

Lemma 2: From the DTMC described in Fig. 4 we obtainthe following steady state probabilities

πn =λn(1− µ)n−1

µn(1− λ)nπ0, n ∈ {1, 2}, (20)

and π0 =λ− µλρ2 − µ

, (21)

where ρ = λ(1−µ)µ(1−λ) , r = λ(1− µ), and s = µ(1− λ).


0 1 2(1− λ)

λ λ(1− µ)

µ(1− λ)µ(1− λ)

1− r − s 1− s

Fig. 4. The DTMC which models the evolution of the queue with replacementat node i.

Proof: See Appendix A. 2

To calculate the average AoI of node i at the destination for thereplacement queue discipline we use (9) and describe an eventsuch thatZj and Tj−1 are conditionally independent. In general,the inter-reception time Zj depends on the system time Tj−1 ofthe previous packet in the system and this complicates the anal-ysis of their joint distribution. We denote by ψj the event thatthe system is empty after the jth successful transmission. Fur-thermore, let ψj be the complementary event that the jth packetleaves behind a system with a packet waiting in the queue. Thenormalized probabilities of these events are given by

P(ψj) =π0

π0 + π1=

µ− λµλ+ µ− λµ

, (22)

P(ψj) =π1

π0 + π1=

λ

λ+ µ− λµ. (23)

Then, the expectations E[ZT ], E[Z], and E[Z2], in (9) can becalculated by conditioning on the events ψj and ψj .

The inter-reception time of the jth packet given that the(j − 1)th packet leaves behind an empty system is given by theconvolution of two independent geometric random variables thatrepresent the interarrival time of update j and the service timeof the same update. Hence,

P{Zj = z|ψj−1} =z−1∑k=1

P{Yj = k}P{Sj = z − k}

=λµ

µ− λ[(1− λ)z−1 − (1− µ)z−1

], (24)

E[Zj |ψj−1] =λ+ µ

λµ, (25)

E[Z2j |ψj−1] =

2λ2 + 2λµ− λ2µ+ 2µ2 − λµ2

λ2µ2. (26)

Moreover, in case there is a packet waiting in the queue thatstarts service as soon as packet (j − 1) completes service, wehave

P{Zj = z|ψj−1} = µ (1− µ)z−1, (27)

E[Zj |ψj−1] =1

µ, (28)

E[Z2j |ψj−1] =

2− µµ2

. (29)

Then, the last two terms in (9) can be obtained as

E[Zj ] = E[Zj |ψj−1]P(ψj−1) + E[Zj |ψj−1]P(ψj−1)

=λ+ µ

λµ

µ− λµ(λ+ µ− λµ)

+1

µ

λ

(λ+ µ− λµ)

=λ2(1− µ) + λ(1− µ)µ+ µ2

λµ(λ+ µ− λµ), (30)

and

E[Z2j ] =E[Z2

j |ψj−1]P(ψj−1) + E[Z2j |ψj−1]P(ψj−1)

=2λ2 + 2λµ− λ2µ+ 2µ2 − λµ2

λ2µ2

µ− λµ(λ+ µ− λµ)

+2− µµ2

λ

(λ+ µ− λµ). (31)

To derive the conditional distributions of service time giventhe events ψj−1 and ψj−1 we note that the (j − 1)th packetleaves behind an empty system if and only if zero arrivals occurwhile it is being served. Then, the conditional distribution ofservice time given the event ψj−1, where fS(·) is the servicetime pmf, is given by

P{Sj−1 = k|ψj−1} =P(ψj−1|Sj−1 = k)fS(k)∑∞k=1 P(ψj−1|Sj−1 = k)fS(k)

=

(k0

)(1− λ)kµ(1− µ)k−1∑∞

k=1

(k0

)(1− λ)kµ(1− µ)k−1

= ((1− λ)(1− µ))k−1(λ+ µ− λµ),(32)

with the resulting conditional expectation

E[Sj−1|ψj−1] =1

λ+ µ− λµ. (33)

For the complementary event ψj−1 the conditional distributionof the service time is given by

P{Sj−1 = k|ψj−1} =P(ψj−1|Sj−1 = k)fS(k)∑∞k=1 P(ψj−1|Sj−1 = k)fS(k)

=(1−

(k0

)(1− λ)k)µ(1− µ)k−1∑∞

k=1(1−(k0

)(1− λ)k)µ(1− µ)k−1

=(1− (1− λ)k)µ(1− µ)k−1(λ+ µ− λµ)

λ,

(34)

with the resulting conditional expectation

E[Sj−1|ψj−1] =λ(1− µ)2 + (2− µ)µ

µ(λ+ µ− λµ)

=1

λ+ µ− λµ+

1

µ− 1. (35)

We proceed with the characterization of the waiting time fortransmitted packets via considering the events of transmission


(tx) or replacement (drop). We consider two possible serverstates of node i, either idle or busy. A packet arrival finds theserver idle with probability P(idle) = π0, due to the BASTAproperty (Bernoulli Arrivals See Time Averages) [54]. A packetarrival finds the server busy with probability P(busy) = 1− π0.This packet will receive service if and only if zero arrivals occurwhile the packet in the server is transmitted. Let R representthe remaining service time of an update, with pmf fR(r), andlet φ be the event that zero arrivals occur during the remainingservice time. For every measurable set A ⊂ [0,∞), we definethe probability

P(φ,R ∈ A) =∑r∈A

P(φ|R = r)fR(r). (36)

Then, the probability of transmission conditioned on the eventthat the server is busy is given by

P(tx|busy) =

∞∑r=0

P(φ|R = r)fR(r)

=

∞∑r=1

(r

0

)(1− λ)rµ(1− µ)r−1

=µ− λµ

λ+ µ− λµ. (37)

As a result,

P(busy, tx) = (1− π0)µ− λµ

λ+ µ− λµ. (38)

The distribution of the waiting time conditioned on the event{busy, tx} is given by

f(w|busy, tx) = f(r|φ) =P(φ|R = r)fR(r)∑∞r=0 P(φ|R = r)fR(r)

=

(r0

)(1− λ)rµ(1− µ)r−1∑∞

r=1

(r0

)(1− λ)rµ(1− µ)r−1

= [(1− λ)(1− µ)]r−1

(λ+ µ− λµ)

= [1− (λ+ µ− λµ)]r−1

(λ+ µ− λµ). (39)

Hence, W conditioned on the event {busy, tx} is geometricallydistributed with parameter (λ+ µ− λµ).

Finally, using (40) the expected value of the waiting time fora transmitted packet is obtained as

E[W |tx] =(1− λ)λ(1− µ)(µ+ λ− 2λµ)

λ2(µ− 1)2 + λ(1− 2µ)µ+ µ2

× 1

(λ+ µ− λµ). (40)

Next, given the conditional expectations of the service time(34) and (36), and the expectation of the waiting time (41), wecalculate the conditional expectations of the system time as fol-

lows

E[Tj−1|ψj−1] = E[Wj−1|ψj−1] + E[Sj−1|ψj−1]

= E[Wj−1] + E[Sj−1|ψj−1]

=1 + (1−λ)λ(1−µ)(λ+µ−2λµ)

λ2(µ−1)2+λ(1−2µ)µ+µ2

(λ+ µ− λµ), (41)

E[Tj−1|ψj−1] = E[Wj−1|ψj−1] + E[Sj−1|ψj−1]

= E[Wj−1] + E[Sj−1|ψj−1]

=(1− λ)λ(1− µ)(µ+ λ− 2λµ)

(λ+ µ− λµ)(λ2(µ− 1)2 + λ(1− 2µ)µ+ µ2)

+1

λ+ µ− λµ+

1

µ− 1. (42)

Utilizing the probabilities (22), (23), the conditional expec-tations of the system time (42), (43), and the conditional ex-pectations of the inter-reception time (25), (28), we calculateE[Tj−1Zj ] as follows

E[Tj−1Zj ] =P(ψj−1)(E[Zj |ψj−1]E[Tj−1|ψj−1])

+ P(ψj−1)(E[Zj |ψj−1]E[Tj−1|ψj−1])

=1

µ2+

1− λλµ

− 1 + λ

(λ+ µ− λµ)2+

1 + 2λ

λ+ µ− λµ

+λ(1− 2µ+ λ(3µ− 2))

λ2(µ− 1)2 + λ(1− 2µ)µ+ µ2. (43)

We refer to the time average rate of packets that enter andremain in the system as the effective rate and define it as

λe = λ(1− pD)

= λ− λ λ2(1− µ)

λ2(1− µ) + λ(1− µ)µ+ µ2, (44)

where pD is the packet dropping probability

pD =λn(1− µ)

µn(1− λ)

(1 +

λ

µ(1− λ)+λn(1− µ)n−1

µn(1− λ)n−1

)−1

(45)

for n = 2.Finally, using (31), (32), (44), (45), and (9), the average age

of information of node i for the replacement discipline is cal-culated as shown in (47). We recall that the analysis providedherein does not consider any coupling between the transmissionqueues but instead focuses on the AoI performance of an inde-pendent queue. Such a step would require knowing the station-ary probability distribution of the joint queue length process.We proceed in the next section by detailing the three proposedaccess policies and evaluating them through simulations.

IV. SIMULATION RESULTS

The objective considered in this paper is to minimize the timeaverage AoI over all policies and all nodes. In that direction,we first investigate all policies without the effect of channel


∆replacement =1

λ2(1− µ) + λ(1− µ)µ+ µ2

(λµ(λ+ µ− λµ)

(λ2(1− µ) + λ(1− µ)µ+ µ2

2λµ(λ+ µ− λµ)+

λ(λ(3µ− 2)− 2µ+ 1)

λ2(µ− 1)2 + λµ(1− 2µ) + µ2

+λ3(µ− 2)(µ− 1) + λ2(µ− 2)(µ− 1)µ+ λµ2(2− 3µ) + 2µ3

2λ2µ2(λ+ µ− λµ)+

1− λλµ

+2λ+ 1

λ+ µ− λµ−

λ+ 1

(λ+ µ− λµ)2+

1

µ2

)). (46)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0

5

10

15

20

25

Fig. 5. Average age per source vs. the arrival rate λi for the round-robinscheduler without packet management at the transmission queues.

fading through simulations. We develop a MATLAB-based be-havioural simulator where each case runs for 106 timeslots.

A. Round-robin Scheduled Access

In the round-robin scheduler nodes take turns to transmit theirstatus updates. If there is no packet at the ith queue waiting fortransmission then the assigned time slot to source i is wastedwith no transmission taking place. Round-robin is a simplescheduler that does not require dynamic coordination but comeswith a throughput loss. Assuming a fixed scheduling interval,each node is assigned a unique time slot index.

In Fig. 5 the average AoI per source is shown as a function ofthe arrival rate per source without any packet management, forλ1 = · · · = λN and success probability 1, at the destination.We observe that the AoI tends to infinity as the arrival rate tendsto 1/N . This is due to the violation of the stability conditionsfor the queues implying infinite queueing delay.

In Fig. 6 the average AoI per source is shown as a functionof the number of source nodes N with a replacement queue, forλ1 = · · · = λN and success probability 1, at the destination. Inthis case, the AoI is a monotonically decreasing function of thearrival rate. Moreover, we note that the AoI increases linearlywith the number of source nodes N . The average AoI for theround-robin scheduler with the replacement queue discipline islower bounded by (N + 3)/2, where N the number of sourcenodes in the system, i.e.,

∆i ≥N + 3

2, ∀i ∈ {1, · · ·, N}, λi ∈ (0, 1). (47)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0

2

4

6

8

10

12

14

Fig. 6. Average age per source vs. the number of sourcesN for the round-robinscheduler with the replacement queue discipline.

B. Work-conserving Scheduled Access

The work-conserving scheduler makes probabilistic decisionsamong the nodes that have a packet at the transmission queueat the same time slot. Specifically, source i is assigned thegiven time slot with probability 1/N , where N is the numberof sources that have a packet available for transmission. A timeslot is wasted with no transmission taking place only when wehave an empty system.

In Fig. 7 the average AoI per source is shown as a functionof the number of source nodes N with a replacement queue dis-cipline, for λ1 = · · · = λN and success probability 1, at thedestination. With solid line we plot the work-conserving sched-uler and with dashed line the round-robin scheduler. The AoI ofsource i for the work-conserving scheduler is a monotonicallydecreasing function of the arrival rate λi, similar to the round-robin scheduler. Moreover, we observe that as λi decreases, thegap between the performance of the work-conserving schedulerand the round-robin scheduler increases. For λi = 1 when thereis always a packet available for transmission the performanceof the two schedulers with respect to the AoI metric coincides.Therefore, the average AoI for the work-conserving schedulerwith the replacement queue discipline is also lower bounded by(N + 3)/2 where N the number of source nodes in the system,i.e.,

∆i ≥N + 3

2, ∀i ∈ {1, · · ·, N}, λi ∈ (0, 1). (48)


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0

2

4

6

8

10

12

14

Fig. 7. Average age per source vs. the number of sources N for the work-conserving scheduler (solid lines) with the replacement queue discipline.The round-robin scheduler is depicted with dashed lines.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0

5

10

15

20

25

Fig. 8. Average age per source vs. the arrival rate λi for the random accesswithout packet management at the transmission queues.

C. Random Access

In the slotted random access policy, at each time slot, nodei attempts to transmit the packet at the head of the queue withprobability qi, provided that the queue is not empty.

In Fig. 8 the average AoI per source is shown as a function ofthe arrival rate per source without any packet management, forλ1 = · · · = λN , q = q1 = · · · = qN = 0.5, and a collisionchannel with success probability Nq(1− q)N−1, at the destina-tion. We observe that the AoI tends to infinity as the arrival ratetends to (1/N) ∗ q. This is due to the violation of the stabilityconditions for the queues implying infinite queueing delay.

In Fig. 9 the average AoI per source is shown as a functionof the number of source nodes N with a replacement queue, forλ1 = · · · = λN , q = q1 = · · · = qN = 0.5, and a collisionchannel with success probability Nq(1 − q)N−1, at the desti-nation. We observe that the arrival rate that minimizes the AoIchanges depending on the number of sources N . Specifically,

1 1.5 2 2.5 3 3.5 4 4.5 5

0

5

10

15

20

25

30

35

Fig. 9. Average age per source vs. the arrival rate λi for the random access withthe replacement queue discipline.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

2

4

6

8

10

12

14

16

18

20

Fig. 10. Average age vs. the access probability qi for the random access withthe replacement queue discipline and N = 2.

for small values of the arrival rate λi it is preferable to havemore source nodes transmitting, while for large values of the ar-rival rate λi it is preferable to have few source nodes. The AoIof source i is a monotonically decreasing function of the arrivalrate λi for N ∈ {1, 2}.

In Fig. 10 the average AoI per source is shown as a function ofthe access probability qi of source i with a replacement queue,for N = 2, λ1 = λ2, and a collision channel, at the destination.In this setup, we can find the optimal access probability qi forvarious arrival rates λi and number of nodes N . It is interestingto see that when qi is small it is better to have a large arrivalrate in order to guarantee that there will be packets availablefor transmission. On the other hand, for large qi a small rate isbeneficial since the absence of packets reduces the collisions.

V. FADING AND NETWORK PATH DIVERSITY

In this section, we consider the effect of the success probabil-ity of a packet erasure model and the effect of the network path


λ2s2 d

λ1s1

λNsN

µ1

µN

µ2

Access point

k

Fig. 11. Status updates over a multiaccess network with out- of-order recep-tions.

diversity on the system, and present how the different parame-ters affect the system performance.

In particular, we assume that the node considered until now asthe destination is an access point (AP). Packets are sent throughwireless channels to the AP and then from the AP they are trans-mitted through an error free network to the final destination, asshown in Fig. 11. After the AP we consider a process that cap-tures the network delay imposed to packets. This is a simplifiedmodel of the random delay experienced by a packet after its de-parture from the AP to the final destination d. This process canmodel several cases, such as the delay for contending with otherpackets in the reception queue, multiple hops, or the process-ing time at the receiver. We model the availability of resourcesand the network path diversity by assuming an infinite numberof servers at the AP. The network delay process follows a geo-metric distribution with mean 1/k, for 0 < k < 1, and it causespackets to arrive at the destination d out of order.

The network delay process can cause out of order receptionof packets at the destination d. We define an informative packetas a packet that carries the newest information compared to thepackets of the same source arriving at the destination prior to it.A packet j is said to be obsolete if there is at least one packetwith k ≥ 1 of the same source generated after j, such that t′j >t′j+k. An informative packet is one that is not rendered obsolete.Obsolete packets correspond to waste of resources since theydo not provide fresh information to the destination. Thus, itis meaningful to minimize the percentage of obsolete packetsamong the transmitted packets.

In Fig. 12 the average AoI per source is shown as a function ofthe arrival rate per source node with a replacement queue, for theround-robin scheduler, λ1 = · · · = λN , and success probabilityp1 = · · · = pN , at the AP. We see that as pi decreases, the gapbetween the performance of the system for N = 2 and N = 3increases. In other words, under good channel conditions addingmore source nodes will degrade the AoI performance less com-pared to the case where the channel conditions are weak.

In Fig. 13 we compare the effect of the parameters pi and kon the AoI objective. Recall that the network delay process fol-lows a geometric distribution with mean 1/k. The average AoIper source is shown as a function of the arrival rate per sourcenode with a replacement queue, for the round-robin scheduler,N = 2, and λ1 = λ2. For the different values of the successprobability pi the AoI is measured at the AP. For the different

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

Fig. 12. Average age per source vs. the arrival rate λi for the round-robinscheduler with the replacement queue discipline. The success probabilityof the ith node is pi.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

2

4

6

8

10

12

14

16

18

20

Fig. 13. Average age per source vs. the arrival rate λi for the round-robinscheduler with the replacement queue discipline, for N = 2. The successprobability of the ith node is pi and the network delay parameter is k.

values of the parameter k the AoI is measured at the destinationd, assuming that the transmission to the AP is instantaneous enderror-free. We observe that the effect of the parameter k differsfrom the effect of the parameter pi. This is due to the fact thata failure in transmission corresponds not only to a wasted timeslot but also to a wasted turn for the source.

In Fig. 14 the number of obsolete packets is shown as a func-tion of the network delay parameter k for the round-robin sched-uler, N = 2, λ1 = λ2, and success probability 1, at the desti-nation d. As expected, increasing the arrival rate at the sourcenodes results in more packets that are rendered obsolete. Hence,there is a tradeoff between the AoI performance and the numberof wasted resources in terms of obsolete packets.


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

210

4

Fig. 14. Obsolete packets vs. the network delay parameter k for the round-robin scheduler with the replacement queue discipline, for N = 2. Thetime horizon is 100000 time slots.

VI. SUMMARY

In this work, we have focused on the AoI performance of anetwork consisting of N source nodes communicating with acommon receiver. Each source node has a buffer of infinite ca-pacity to store incoming bursty traffic in the form of packetswhich should keep the receiver timely updated. We have consid-ered two different queue disciplines at the transmission queues,with and without packet management, and we have derived an-alytical expressions for the AoI for both cases. We have investi-gated three different policies to access the common medium (i)round-robin scheduler (ii) work-conserving scheduler (iii) ran-dom access. The work-conserving scheduler outperforms theround-robin scheduler. For the case of the random access oneshould optimize the access probabilities in connection to the ar-rival rates per source and the number of source nodes in thesystem. Moreover, we have considered the effect of the successprobability of a packet erasure model and the effect of networkpath diversity, on the AoI performance. The presented simula-tion results provide guidelines for the design of the system.

APPENDIX APROOF OF LEMMA 2

Given the DTMC described in Fig. 4 we define r = λ(1− µ)and s = µ(1− λ) and obtain the following balance equations:

λπ0 = µ(1− λ)π1 ⇔ π1 =λ

µ(1− λ)π0,

π1 = λπ0 + (1− r − s)π1 + sπ2 ⇔ π2 =λ2(1− µ)

µ2(1− λ)2π0.

Summarizing, for n ∈ {1, 2} we have that

πn =λn(1− µ)n−1

µn(1− λ)nπ0.

Moreover, we know that

π0 +λ

µ(1− λ)π0 +

λ2(1− µ)

µ2(1− λ)2π0 = 1.

Hence, the probability that the queue is empty is given by

π0 =λ− µ

λ(λ(1−µ)µ(1−λ)

)2

− µ.

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Antzela Kosta received the diploma degree in elec-tronic and computer engineering from the Techni-cal University of Crete (TUC), Chania, Greece, in2015. She is currently pursuing the Ph.D. degreeat the Department of Science and Technology (ITN),Linköping University (LiU), Sweden, as a MarieCurie Fellow (IAPP). From January 2015 to July2015, she was a Graduate Research Assistant with theTelecommunication Systems Institute (TSI), Techni-cal University of Crete, Greece. Since May 2017, shehas been a Research Intern with Ericsson Research,

Linköping, Sweden. Her research interests include communication theory,queueing theory, age-of-information, stochastic optimization, and radio resourcemanagement in wireless networks.

Nikolaos Pappas received the B.Sc. degree in com-puter science, the B.Sc. degree in mathematics, theM.Sc. degree in computer science, and the Ph.D. de-gree in computer science from the University of Crete,Greece, in 2005, 2012, 2007, and 2012, respectively.From 2005 to 2012, he was a Graduate Research As-sistant with the Telecommunications and NetworksLaboratory, Institute of Computer Science, Founda-tion for Research and Technology-Hellas, and a Vis-iting Scholar with the Institute of Systems Research,University of Maryland at College Park, College Park,

MD, USA. From 2012 to 2014, he was a Post-Doctoral Researcher with the De-partment of Telecommunications, Supélec, France. Since 2014, he has been withLinköping University, Norrköping, Sweden, as a Marie Curie Fellow (IAPP). Heis currently an Associate Professor in mobile telecommunications with the De-partment of Science and Technology, Linköping University. His main researchinterests include the field of wireless communication networks with emphasis onthe stability analysis, energy harvesting networks, network-level cooperation,age-of-information, network coding, and stochastic geometry. From 2013 to2018, he was an Editor of the IEEE COMMUNICATIONS LETTERS. He is cur-rently an Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS andthe IEEE/KICS JOURNAL OF COMMUNICATIONS AND NETWORKS.


Anthony Ephremides received the Ph.D. degree inelectrical engineering from Princeton University in1971. Since 1971, he has been with the Universityof Maryland, College Park, MD, USA. He holds theCynthia Kim Professorship of information technologyat the Electrical and Computer Engineering Depart-ment, University of Maryland, where he is a Distin-guished University Professor and has a joint appoint-ment at the Institute for Systems Research, of whichhe was among the founding members in 1986. He hasauthored several hundred papers, conference presen-

tations, and patents. His research interests lie in the areas of communicationsystems and networks and all related disciplines, such as information theory,control and optimization, satellite systems, queueing models, signal processing,and so on. He is especially interested in wireless networks, energy efficientsystems, and the new notion of age of information.

Vangelis Angelakis received the Ph.D. degree fromthe Department of Computer Science, University ofCrete, Greece, in 2008. In 2015, he received the Do-cent from the Department of Science and Technology(ITN), Linköping University (LiU), Norrköping, Swe-den. He is currently an Associate Professor with ITN,LiU. He has authored over 60 papers in internationaljournals and peer-reviewed conferences. His researchinterests revolve around the design of telecommuni-cation systems and networks resources optimizationwith a focus on IoT and smart city applications. He

has held multiple guest posts both in the industry and academia, primarilythrough EU funded projects, in US, EU, and China. He is coordinating the EUFP7 MSCA IAPP Project (SOrBet), and is a Principal Investigator for LiU in theEU FP7 SMARTCITIES RERUM Project and the EU Horizon 2020 MSCA ITNWiVi-2020 Project. He is also acting as the Project Manager for two more EU-funded projects coordinated by the Mobile Telecommunications Group and hasbeen with the OPTICWISE COST Action Management Committee. He is cur-rently editing a Springer book on smart cities, due in 2016, and co-organizingthe IEEE GLOBECOM15 workshop in optimizing heterogeneous networkingtechnologies for the Internet of Things. He serves as an Associate Editor forthe IEEE/KICS JOURNAL OF COMMUNICATIONS AND NETWORKS. Hehas been serving as an organizer and technical program committee member in awide range of international conferences and workshops.

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Age of Information Performance of Multiaccess Strategies ...

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